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Theorem iscmgmALT 47064
Description: The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b 𝐵 = (Base‘𝑀)
ismgmALT.o = (+g𝑀)
Assertion
Ref Expression
iscmgmALT (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))

Proof of Theorem iscmgmALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6891 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 5161 . . 3 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ (+g𝑀) comLaw (Base‘𝑀)))
4 ismgmALT.o . . . 4 = (+g𝑀)
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5breq12i 5157 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6bitr4di 289 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-cmgm2 47060 . 2 CMgmALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3686 1 (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wcel 2105   class class class wbr 5148  cfv 6543  Basecbs 17151  +gcplusg 17204   comLaw ccomlaw 47025  MgmALTcmgm2 47055  CMgmALTccmgm2 47056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-cmgm2 47060
This theorem is referenced by: (None)
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